A086369 -id:A086369 - cp's OEIS frontend

A385209 Least k such that A086369(k) = n.

1, 2, 3, 4, 9, 6, 15, 16, 81, 12, 45, 64, 729, 24, 105, 36, 225, 48, 405, 1024, 59049, 60, 315, 4096, 531441, 192, 3645, 144, 2025, 120, 945, 65536, 43046721, 180, 1575, 262144, 387420489, 240, 2835, 576, 18225, 3072, 295245, 4194304, 31381059609, 360, 3465, 1296

Offset: 1

Miles Englezou, Jun 21 2025

nonn

A086369(n) is also the cardinality of the set containing the divisors d of n and those 0 < m < n satisfying m + d = n.

			a(4) = 4 since 4 is the least k such that A086369(k) = 4.
a(13) = 729 since 729 is the least k such that A086369(k) = 13.

Cf. A086369, A385202.

a(n,k) = my(f); (f(n) = if(n%2==1, 1+2*sumdiv(n,d,d

a(2*p-1) = 3^(p-1) for prime p. - Jinyuan Wang, Jun 30 2025

a(45)-a(48) from Jinyuan Wang, Jun 30 2025

A086374 Number of factors over Q in the factorization of T_n(x) + 1 where T_n(x) is the Chebyshev polynomial of the first kind.

Original entry on oeis.org

1, 2, 3, 2, 3, 4, 3, 2, 5, 4, 3, 4, 3, 4, 7, 2, 3, 6, 3, 4, 7, 4, 3, 4, 5, 4, 7, 4, 3, 8, 3, 2, 7, 4, 7, 6, 3, 4, 7, 4, 3, 8, 3, 4, 11, 4, 3, 4, 5, 6, 7, 4, 3, 8, 7, 4, 7, 4, 3, 8, 3, 4, 11, 2, 7, 8, 3, 4, 7, 8, 3, 6, 3, 4, 11, 4, 7, 8, 3, 4, 9, 4, 3, 8, 7, 4, 7, 4, 3, 12, 7, 4, 7, 4, 7, 4, 3, 6, 11, 6, 3, 8, 3

Offset: 1

Yuval Dekel (dekelyuval(AT)hotmail.com), Sep 06 2003

			a(6) = 4 because T_6(x)+1 = 32x^6-48x^4+18x^2 = x^2*(4x^2-3)^2.

Antti Karttunen, Table of n, a(n) for n = 1..2049

Cf. A001227, A086369.

p2 = 1; p1 = x; for (n = 1, 103, p = 2*x*p1 - p2; f = factor(p1 + 1); print(sum(i = 1, matsize(f)[1], f[i, 2]), " "); p2 = p1; p1 = p); \\ David Wasserman, Mar 03 2005

A086374(n) = {vecsum(factor(polchebyshev(n, 1, x)+1)[, 2])}; \\ Antti Karttunen, Sep 27 2018, after Andrew Howroyd's program for A086369

If p is an odd prime then a(p) = 3.

More terms from David Wasserman, Mar 03 2005

A360999 Number of tilings of an n X 2 rectangle by integer-sided rectangular pieces that cannot be rearranged to produce a different tiling of the rectangle (including rotations and reflections of the original tiling).

Original entry on oeis.org

2, 2, 3, 4, 3, 6, 3, 6, 5, 6, 3, 10, 3, 6, 7, 8, 3, 10, 3, 10, 7, 6, 3, 14, 5, 6, 7, 10, 3, 14, 3, 10, 7, 6, 7, 16, 3, 6, 7, 14, 3, 14, 3, 10, 11, 6, 3, 18, 5, 10, 7, 10, 3, 14, 7, 14, 7, 6, 3, 22, 3, 6, 11, 12, 7, 14, 3, 10, 7, 14, 3, 22, 3, 6, 11, 10, 7, 14

Offset: 1

Pontus von Brömssen, Feb 28 2023

nonn

Second column of A360998.
Essentially the same as A086369.
Cf. A000005, A000035, A114003, A360631, A361004.

a(n) = 2*A000005(n) - 1 - [n even] = A114003(n) + A000035(n) - 1 for n >= 2.

A385202 Irregular triangle read by rows: let S be an ordered set of nondivisors of n such that a and b belong to S if a + b = n. T(n,k) is the k-th member of S. If S is empty, T(n,k) = 0.

Original entry on oeis.org

0, 0, 0, 0, 2, 3, 0, 2, 3, 4, 5, 3, 5, 2, 4, 5, 7, 3, 4, 6, 7, 2, 3, 4, 5, 6, 7, 8, 9, 5, 7, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 3, 4, 5, 6, 8, 9, 10, 11, 2, 4, 6, 7, 8, 9, 11, 13, 3, 5, 6, 7, 9, 10, 11, 13, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 4, 5, 7, 8, 10, 11, 13, 14

Offset: 1

Miles Englezou, Jun 21 2025

			 n | Triangle begins:
---+-----------------
 1 | {}
 2 | {}
 3 | {}
 4 | {}
 5 | {2, 3}
 6 | {}
 7 | {2, 3, 4, 5}
 8 | {3, 5}
 9 | {2, 4, 5, 7}
10 | {3, 4, 6, 7}
11 | {2, 3, 4, 5, 6, 7, 8, 9}
12 | {5, 7}
13 | {2, 3, 4, 5, 6, 7, 8, 9, 10, 11}
14 | {3, 4, 5, 6, 8, 9, 10, 11}
15 | {2, 4, 6, 7, 8, 9, 11, 13}
16 | {3, 5, 6, 7, 9, 10, 11, 13}
17 | {2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15}
18 | {4, 5, 7, 8, 10, 11, 13, 14}
19 | {2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17}
20 | {3, 6, 7, 8, 9, 11, 12, 13, 14, 17}
21 | {2, 4, 5, 6, 8, 9, 10, 11, 12, 13, 15, 16, 17, 19}
22 | {3, 4, 5, 6, 7, 8, 9, 10, 12, 13, 14, 15, 16, 17, 18, 19}
23 | {2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21}
24 | {5, 7, 9, 10, 11, 13, 14, 15, 17, 19}
25 | {2, 3, 4, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 21, 22, 23}

Cf. A086369.

T(n,k) = my(S); S = select(x -> setsearch(divisors(n), x)==0 && setsearch(divisors(n), n-x)==0, [1..n]); if(k <= #S, S[k], 0) \\ function made to output 0 if k exceeds the size of S to avoid breaking

n = T(n, m) + T(n, k-(m-1)), 1 <= m <= k, for every row of length k.

S defined as in the name, n - |S| = A086369(n).

cp's OEIS Frontend

A385209 Least k such that A086369(k) = n.

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A086374 Number of factors over Q in the factorization of T_n(x) + 1 where T_n(x) is the Chebyshev polynomial of the first kind.

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PARI

PARI

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Extensions

A360999 Number of tilings of an n X 2 rectangle by integer-sided rectangular pieces that cannot be rearranged to produce a different tiling of the rectangle (including rotations and reflections of the original tiling).

Views

Author

Keywords

Crossrefs

Formula

A385202 Irregular triangle read by rows: let S be an ordered set of nondivisors of n such that a and b belong to S if a + b = n. T(n,k) is the k-th member of S. If S is empty, T(n,k) = 0.

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Examples

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PARI

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